Monadic and cylindric expansions of bounded implication algebras

Implication algebras were introduced by Abbott as algebraic models of the operation of Boolean implication in the classical propositional calculus. In this work, we study additional operators and constants on bounded implication algebras by introducing monadic and cylindric implication algebras. It is demonstrated that the category $\mathbf{M}\mathbf{I}\mathbf{A}$ of monadic implication algebras is isomorphic to the category $\mathbf{M}\mathbf{B}\mathbf{A}$ of monadic Boolean algebras and moreover, that the category $\mathbf{C}\mathbf{I}\mathbf{A}$ of $I$-dimensional cylindric implication algebras is isomorphic to the category $\mathbf{C}\mathbf{B}\mathbf{A}$ of $I$-dimensional cylindric Boolean algebras. As an application of the obtained categorical isomorphisms, we provide spectral duality results for $I$-dimensional cylindric implication algebras along the lines of Bezhanishvili and Holliday’s spectral duality for Boolean algebras combined with McDonald’s extension of their duality to monadic and $I$-dimensional cylindric Boolean algebras.